Old Final Exam

Michael Johnson Dept of Math and Comp Sci

CS 543 Final Exam

Date: Jan 11, 2004 Duration: 120 minutes



x3 − x 2 + x − 1 if 0 ≤ x < 1 1. Let s(x) = , and let Ξ = {0, 1, 5}. (x − 1)3 + 2(x − 1)2 + 2(x − 1) if 1 ≤ x ≤ 5 For each statement, decide whether it is True or False and justify your answer. a) s ∈ P3,Ξ b) s ∈ P4,Ξ c) s ∈ S3,Ξ d) s ∈ S4,Ξ e) s ∈ Π3 | [1,5]

2. Explain how one would find a spline s of degree 3 which satisfies the following: s(1) = 5 s00 (1) = 0

s(3) = 2

s(4) = 3 s0 (4) = 2

Be sure to include a) the form of s, in terms of certain unknown coefficients, and b) the equations, in matrix form, which determine the unknown coefficients. 3. Let f ∈ C 4 (R) and suppose we are given the values {f (j/4)}j∈Z . a) Explain how one would construct a function s ∈ C 4 (R) which approximates f b) How would one evaluate this function s at a point x? c) How would one evaluate the fourth derivative of s at a point x? d) State the relevant error estimate concerning the error f − s. 4. Let Φ(x, y) = ψ1 (x)ψ1 (y), where ψ1 is the centered cardinal B-spline of degree 1. Find s(1/2, −1/2) if s = Φ ∗0 f , where f (x, y) = x2 y 3 . 5. Suppose f, g ∈ Cc (R) are functions with supp f ⊂ [0, 7] and supp g ⊂ [0, 9]. a) Prove that (f ∗ g)(x) = 0 whenever x > 16. b) Prove that F (f ∗ g) = fbgb.

6. Let f (x) = e−2|x| and g(x) = e−|x+3| , x ∈ R. a) Find the Fourier transform of f ∗ g. b) Let ζ = f ∗ g. Show that ζ(· + τ ) is positive definite if τ =?.

7. Explain how one would construct a function s ∈ C(R2 ) which satisfies s(1, 2) =

√

2,

s(−1, 4) = π,

s(2, 5) =

√

3,

s(3, −2) = 1/3.

Be sure to include a) the form of s, in terms of certain unknown coefficients, and b) the equations, in matrix form, which determine the unknown coefficients. 8. Let y = [1, 0, 2, 1], and let Y = [Y0 , Y1 , Y2 , Y3 ] ∈ C4 be the 4-point discrete Fourier transform of y. Use the Fast Fourier Transform algorithm to find Y2 .